Radial strength functions

Effect of Diatom Stretching Dependence. The features of the poten-tial energy surface most central to a discussion of its effect on the predissociation process are not the individual radial strength functions V j((R), but rather the vibrational matrix elements (integrated over the diatom bond length) of the full potential... 

If the basis set contains N channel functions of a particular symmetry, a single solution of the coupled equations 1.42 is a vector i f(r) made up of N radial strength functions tlr (r) for j = 1 to N. However, at any energy there are N linearly independent solution veetors that satisfy the boundary conditions (1.37) at short range, and it is usually not possible to select a single one of them that is physically relevant until long-range boundary conditions are applied. In computational terms it is therefore usually necessary to solve for all N solutions simultaneously to obtain aaN x N wavefunction matrix (r) with elements rlfy (r). 

The radial strength functions [fg(r) are in general complex. However, it is almost always possible to transform the problem to work with real functions and to express the complex solutions, when needed, in terms of these. For both open and closed channels, the boundary conditions require that l (r) 0 at short range. However, at long range the scattering boundary conditions corresponding to real solutions are... 

In the central field approximation, when radial wave functions not depending on term are usually employed, the line strengths of any transition may be represented as a product of one radial integral and of a number of 3n./-coefficients, one-electron submatrix elements of standard operators (C(fc) and/or L(1 S(1)), CFP (if the number of electrons in open shells changes) and appropriate algebraic multipliers. It is usually assumed that the radial integral does not depend on the quantum numbers of the vec-... 

The literature on high temperature fused salts is extensive [72-74], and we make only the most cursory review here. The first and most obvious statement about fused salts is that they are fundamentally different than molecular liquids, in that they retain a substantial degree of order on melting. The strength of interion Coulomb interactions mandates that ions be surrounded by counterions, and so maintain the most uniform possible charge distribution throughout the liquid. This expectation is bom out by X-ray [75, 76] and neutron diffraction [76, 77] experiments, which indicate that molten salts retain much of their solid-state structure in the liquid state a representative radial distribution function for a molten salt is given in Fig. 2. 

The strength of the water-metal interaction together with the surface corrugation gives rise to much more drastic changes in water structure than the ones observed in computer simulations of water near smooth nonmetallic surfaces. Structure in the liquid state is usually characterized by pair correlation functions (PCFs). Because of the homogeneity and isotropy of the bulk liquid phase, they become simple radial distribution functions (RDFs), which do only depend on the distance between two atoms. Near an interface, the PCF depends not only on the interatomic distance but also on the position of, say the first, atom relative to the interface and the direction of the interatomic distance vector. Hence, considerable changes in the atom-atom PCFs can be expected close to the surface. 

In this context we should mention the work of Lynden-Bell, who showed that if the strength of the HB interaction is decreased relative to the shorter-ranger Lennard-Jones interaction, then the water-like anomalies disappear in three-dimensional water [ 17]. In this case one finds that lowering the HB energy gradually removes the peak in the oxygen-oxygen radial distribution function and makes the microscopic structures look like a normal liquid. 

The association strength is the key property characterizing the association bond and depends on the segment diameter and the hard-sphere radial distribution function (Eq. (10.37)) ... 

Pressure affects the energetic structure of the Ln ion and the energies of the 4f -4f transitions in two ways (1) the nephelauxetic [115] reduction of the Slater integrals and (2) the spin-orbit coupling constant. This effect was broadly discussed by Shen and Holzapfel [116] in the framework of the covalence model. In this framework, the radial-wave functions of the Ln ion, when it is embedded in the lattice, expand compared with those of a free ion due to the penetration of the ligand electrons into the Ln ion space. Penetration increases when pressure decreases the Ln ion-ligand distances. The second effect is related to the pressure-induced increase of the crystal-field strength. 

Figure 4 Structure in Monte Carlo-simulated dispersion (particle diameter 1.190 p.m, ionic strength 0.1 mol m ). The radial distribution function g(r) is plotted against the reduced separation r/

In Figures 14 and 15 we show the radial distribution functions of water surrounding the infinitely dilute He and Ne in comparison to that of the surrounding water (ideal solution) at the two state conditions. Beyond the small differences in strength, there are clear indications that water forms a cavity around these solutes, i.e., the water local density around each solute is much smaller than that found around any water molecule (ideal solution). Moreover, due to the closeness to the water critical conditions, these distribution functions show the characteristic slow-decaying (compressibility driven) tails, from above unity and below unity for... 

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